Abstract
The structure of the set of closed two-sided ideals in aC*-algebraU with identity is described by means of a topology on the set ∂e K of extreme points of the state spaceK ofU. Recent results of Alfsen, Andersen, Combes, Perdrizet, Wils, and others have shown that such a topology can be defined on the set ∂e K of extreme points of an arbitrary compact convex subset of a locally convex Hausdorff topological vector space.
The structure of the set of closed left ideals in aC*-algebraU with identity can also be described by means of a set of subsets of the set ∂e K of extreme points of its state spaceK. Akemann, Giles, and Kummer showed that this formed a more general structure than a topology which was called aq-topology. In this paper it is shown that for a reasonably wide class of compact convex subsetsK of locally convex Hausdorff topological vector spaces such aq-topology can also be defined on ∂e K and that it shares many of the properties of theq-topology defined forC*-algebras. The methods used depend strongly upon recent results of Alfsen and Shultz on the spectral theory of affine functions on compact convex sets.
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Edwards, C.M. The facialQ-topology for compact convex sets. Math. Ann. 230, 123–152 (1977). https://doi.org/10.1007/BF01370659
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DOI: https://doi.org/10.1007/BF01370659